CBSE Class 10 Maths Chapter 13 Surface Areas and Volumes Notes

SURFACE AREA AND VOLUME OF COMBINATIONS

Cone on a Cylinder.

r = radius of cone & cylinder;
h₁ = height of cone
h₂ = height of cylinder
Total Surface area = Curved surface area of cone + Curved surface area of cylinder + area of circular base
= πrl + 2πrh₂ +πr²;
Slant height, l = $\sqrt { { r }^{ 2 }+{ { { h }_{ 1 }^{ 2 } } } }$
Total Volume = Volume of cone + Volume of cylinder
= $\frac { 1 }{ 3 } { \pi r }^{ 2 }{ h }_{ 1 }+{ \pi r }^{ 2 }{ h }_{ 2 }$

Cone on a Hemisphere:

h = height of cone;
l = slant height of cone = $\sqrt { { r }^{ 2 }+{ h }^{ 2 } }$
r = radius of cone and hemisphere
Total Surface area = Curved surface area of cone + Curved surface area of hemisphere = πrl + 2πr²
Volume = Volume of cone + Volume of hemisphere = $\frac { 1 }{ 3 } { \pi r }^{ 2 }h+\frac { 2 }{ 3 } { \pi r }^{ 3 }$

Conical Cavity in a Cylinder

r = radius of cone and cylinder;
h = height of cylinder and conical cavity;
l = Slant height
Total Surface area = Curved surface area of cylinder + Area of bottom face of cylinder + Curved surface area of cone = 2πrh + πr² + πrl
Volume = Volume of cylinder – Volume of cone = ${ \pi r }^{ 2 }h-\frac { 1 }{ 3 } { \pi r }^{ 2 }h=\frac { 2 }{ 3 } { \pi r }^{ 2 }h$

Cones on Either Side of Cylinder.

r = radius of cylinder and cone;
h₁ = height of cylinder
h₂ = height of cones
Slant height of cone, l = $\sqrt { { h }_{ 2 }^{ 2 }+{ r }^{ 2 } }$
Surface area = Curved surface area of 2 cones + Curved surface area of cylinder = 2πrl + 2πrh₁
Volume = 2(Volume of cone) + Volume of cylinder = $\frac { 2 }{ 3 } { \pi r }^{ 2 }{ h }_{ 2 }+{ \pi r }^{ 2 }{ h }_{ 1 }$

Cylinder with Hemispherical Ends.

r = radius of cylinder and hemispherical ends;
h = height of cylinder
Total surface area= Curved surface area of cylinder + Curved surface area of 2 hemispheres = 2πrh + 4πr²
Volume = Volume of cylinder + Volume of 2 hemispheres = ${ \pi r }^{ 2 }h+\frac { 4 }{ 3 } { \pi r }^{ 3 }$

Hemisphere on Cube or Hemispherical Cavity on Cube

a = side of cube;
r = radius of hemisphere.
Surface area = Surface area of cube – Area of hemisphere face + Curved surface area of hemisphere
= 6a² – πr² + 2πr² = 6a² + πr²
Volume = Volume of cube + Volume of hemisphere = ${ a }^{ 3 }+\frac { 4 }{ 3 } { \pi r }^{ 3 }$

Hemispherical Cavity in a Cylinder

r = radius of hemisphere;
h = height of cylinder
Total surface area = Curved surface area of cylinder + Surface area of base + Curved surface area of hemisphere
= 2πrh + πr² + 2πr² = 2πrh + 3πr²
Volume = Volume of cylinder – Volume of hemisphere = ${ \pi r }^{ 2 }h-\frac { 2 }{ 3 } { \pi r }^{ 3 }$